Pacific Journal of Mathematics

ON A PRIME ZETA FUNCTION OF A GRAPH

TAKEHIRO HASEGAWA AND SEIKEN SAITO

Volume 273 No. 1 January 2015 PACIFIC JOURNAL OF MATHEMATICS Vol. 273, No. 1, 2015 dx.doi.org/10.2140/pjm.2015.273.123

ON A PRIME ZETA FUNCTION OF A GRAPH

TAKEHIRO HASEGAWA AND SEIKEN SAITO

In the first half of this paper, we introduce a prime zeta function associated with the Ihara zeta function, and study several properties of this function. In the last half, using results of the first half, we present graph-theoretic analogues to Mertens’ theorems.

1. Introduction

Throughout this paper, we use the notation of[Stark and Terras 1996; Terras 2011] for graph theory and the theory of (Ihara) zeta functions Z X (u) of graphs, and the notation of[Hardy and Wright 2008] and [Titchmarsh 1958; 1986] for the theory of functions and the Riemann zeta function ζ(s). In the analytic theory of the Riemann zeta function, the following theorems are well-known: • Mertens’ first theorem[1874, Equality (5)] (also see[Hardy and Wright 2008, Theorem 425],[Jameson 2003, Theorem 2.6.3], and[Titchmarsh 1986, Equal- ity (3.14.3)]): as x → ∞, X log p = log x + O(1). p p≤x • Mertens’ second theorem[1874, Equality (13)] (also see[Hardy and Wright 2008, Theorem 427],[Jameson 2003, Theorem 2.6.4/Exercise 4, p. 191], and [Titchmarsh 1986, Equality (3.14.5)]): as x → ∞, X 1  1  = log(log x) + B + O p 1 k p≤x log x

for each k ≥ 1, where B1 = 0.26149 ... is the Mertens constant. • Mertens’ third theorem[1874, Equality (15)] (also see[Hardy and Wright 2008, Theorem 429],[Jameson 2003, Exercise 1, p. 96], and[Titchmarsh 1986,

MSC2010: primary 11N45; secondary 05C30, 05C38, 05C50. Keywords: Ihara zeta functions, primes in graphs, Mertens’ theorem.

123 124 TAKEHIRO HASEGAWA AND SEIKENSAITO

Equality (3.15.2)]): as x → ∞,

− Y 1  e γ 1 − ∼ , p x p≤x log

where γ = 0.57721 ... is the Euler–Mascheroni constant. • Prime number theorem (proved by de la Vallée Poussin and Hadamard in 1896; see, e.g.,[Hardy and Wright 2008, Theorem 6],[Jameson 2003, Theorem 3.4.3], and[Titchmarsh 1986, Equality (3.7.1)]): as x → ∞, x π(x) ∼ , log x where π(x) denotes the number of rational prime numbers p less than x, that is,

π(x) := {p : p is a rational prime number with p ≤ x} .

All proofs of the above formulae are related to the Riemann zeta function

Y 1 −1 ζ(s) = 1 − , ps p∈ᏼ where ᏼ denotes the set of all rational prime numbers, that is,

ᏼ := {p ∈ ޚ : p is a rational prime number}, and to the prime zeta function, defined first by Glaisher[1891], X 1 P(s) = . ps p∈ᏼ

In graph theory, there exists an analogue of the Riemann zeta function, the so-called (Ihara) zeta function Z X (u) of a graph X (see[Ihara 1966]). Therefore, studying graph-theoretic analogues of these theorems is very interesting. Indeed, Terras and coworkers gave an analogue of the prime number theorem (see Theo- rem 2.10 in[Horton et al. 2006], and also Theorem 10.1 in[Terras 2011]): If 1X divides n, then, as n → ∞, 1 ∼ X πX (n) n , n · RX and otherwise πX (n) ∼ 0. (For the definitions of πX (n) and RX , see this sec- tion, and for that of 1X , see Section 3.) This is called the graph-theoretic prime number theorem. ONAPRIMEZETAFUNCTIONOFAGRAPH 125

In this paper, we define a prime zeta function of a graph, and investigate several properties of this function. In particular, we show that this has a natural bound- ary. Moreover, by using this function, we present graph-theoretic analogues of Mertens’ theorems. We shall note a relation between previous works and our works. A zeta function of a graph can be specialized from a dynamical zeta function for a flow (see Chapter 4 in[Terras 2011]), and dynamical-systemic analogues to the above formulae are already known (see, e.g.,[Sharp 1991] for Mertens’ theorems, and[Parry 1983; Parry and Pollicott 1983] for a prime number theorem). In that sense, our statements for Mertens’ theorems are not new (see Remark 17). However, since our proofs are graph-theoretic and elementary, they are completely different from previous proofs. In this section, we first recall the notation for graph theory and zeta functions of graphs, define a prime zeta function of a graph, and finally state the main theorem. Now we recall the notation of graph theory. Throughout this paper, we always assume that X is a finite, connected, non-cycle and undirected graph without degree- one vertices. Let X be a graph with vertex set V , with ν := |V |, and edge set E, with  := |E|. Simply, such a graph X is denoted by X := (V, E). Note that  is the number of edges of X. An oriented edge (or an arc) a from a vertex u to a vertex v is denoted by a = (u, v), and the inverse of a is denoted by a−1 = (v, u). The origin and terminus of a are denoted by o(a) and t(a), respectively. We can now orient the edges of X, and label the edges as follows:

E = { = −1 = −1 = −1} E e1, e2,..., e, e+1 e1 , e+2 e2 ,..., e2 e .

A path C = a1 ··· as, where the ai are oriented edges, is said to have a backtrack = −1 = −1 (resp. tail) if a j+1 a j for some j (resp. as a1 ), and a path C is called a cycle (or a closed path) if o(a1) = t(as). The length `(C) of a path C = a1 ··· as is defined by `(C) = s. A cycle C is called prime (or primitive) if it satisfies the following:

• C does not have backtracks or a tail; • no cycle D exists such that C = D f for some f > 1.

The equivalence class [C] of a cycle C = a1 ··· as is defined as the set of cycles

[C] := {a1a2 ··· as−1as, a2 ··· as−1asa1,..., asa1a2 ··· as−1}, and an equivalence class [P] of a prime cycle P is called a prime in the graph X. Throughout this paper, we denote a prime by the symbol [P] . Two cycles C1 and C2 are called equivalent if C2 ∈ [C1]. Note that if [C1] = [C2], then `(C1) = `(C2), and thus u`(C1) = u`(C2). 126 TAKEHIRO HASEGAWA AND SEIKENSAITO

Next, we recall the zeta function of a graph X = (V = {v1, . . . , vν}, E), and we define a prime zeta function associated with it. Let u be a complex variable, and let f X (u) denote 2 f X (u) := det(Iν − Au + Qu ), where Iν is the ν × ν identity matrix, A is the adjacency matrix of X (see Defini- tion 2.1 in[Terras 2011]), and

Q = diag(deg(v1) − 1,..., deg(vν) − 1).

Let πX (n) denote

π(n) = πX (n) := {[P]:[P] is a prime in X with `(P) = n} . Throughout this paper, we fix an arbitrary real number t >1 (that is, log t >0), and we set u = t−s. The (Ihara) zeta function of X (see Definition 2.2 and Theorem 2.5 in[Terras 2011]) and the prime zeta function of X are defined as follows:

Y `(P) −1 1 −s Z X (u) : = (1 − u ) = , ᐆX (s) := Z X (t ), (1 − u2)−ν f (u) [P] X ∞ X `(P) X n −s PX (u) : = u = πX (n)u , ᏼX (s) := PX (t ), [P] n=1 with |u| sufficiently small, where [P] runs through all primes in X. In this paper, we do not distinguish between the two functions Z X (u) and ᐆX (s), or between PX (u) and ᏼX (s). The right-hand side of the first equality is called the Ihara–Bass formula (see[Bass 1992]). Note that, owing to our assumption for X, the zeta function Z X (u) is expressible like that. Note that, for two finite connected graphs X1 and X2 without degree-one vertices, = = PX1 (u) PX2 (u) if and only if Z X1 (u) Z X2 (u) (see Proposition 7 in[Storm 2010]). Let ∞ [ n T := Tn and Tn := {u ∈ ރ : f X (u ) = 0} n=1 n n be the zeroes of the f X (u ). Note that the elements of Tn are poles of Z X (u ). The radius of convergence of Z X (u) is denoted by RX . Note that 0 < RX < 1 since X is a non-cycle graph (see, e.g.,[Terras 2011, p. 197]). It follows from the graph-theoretic prime number theorem (see Theorem 10.1 in[Terras 2011]) that the radius of convergence of the other function PX (u) is also equal to RX . Note that the point u = RX is a singularity of PX (u), and that

PX (u) ∼ −log(RX − u) ONAPRIMEZETAFUNCTIONOFAGRAPH 127 as u ↑ RX , which is similar to

P(s) ∼ −log(s − 1)

P s as s ↓ 1 (see, e.g.,[Fröberg 1968]), where P(s) = p 1/p denotes the prime zeta function associated with the Riemann zeta function. Euclid proved that the number of primes p is infinite. Euler showed that the P prime zeta function p 1/p diverges, and as an application he proved the infinitude of primes. In graph theory, it is also well known that the number of primes [P] in X is infinite. We can give another proof “à la Euler” for this fact since u = RX is a singularity of PX (u). Our main theorem is:

Main Theorem. Suppose that X = (V, E) is a finite, connected and non-cycle graph without degree-one vertices.

(1) Let µ(n) denote the Möbius function. If |u| < RX , then

∞ X µ(n) P (u) = log Z (un). X n X n=1 Furthermore, the right-hand side is absolutely convergent for u satisfying |u| < 1 and u ∈/ T , and so PX (u) has an analytic extension to the region {u ∈ ރ : |u| < 1}\ T.

(2) The imaginary axis Re(s) = 0 is a natural boundary for the function ᏼX (s), that is, every point on this line can be realized as a limit point of singularities of ᏼX (s). (3)( Graph-theoretic Mertens’ first theorem) As N → ∞,

X n n · πX (n)RX = N + O(1). n≤N

(4) (Graph-theoretic Mertens’ second theorem) There exists a constant BX such that, as N → ∞, X  1  π (n)Rn = log N + B + O . X X X N n≤N

(5) (Graph-theoretic Mertens’ third theorem) Let γ = 0.57721 ... denote the Euler–Mascheroni constant. As N → ∞,

−γ Y `(P) e 1 (1 − RX ) ∼ · , CX N `(P)≤N 128 TAKEHIRO HASEGAWA AND SEIKENSAITO

where 1 C = − X 2 −ν 0 (1 − RX ) RX f X (RX ) (for the definition, in detail, see Section 3 in this paper). The contents of this paper are as follows. In the next section, we prove the first two claims in the main theorem, that is, several properties of PX (u). In Section 3, we prove the remaining claims in the main theorem, namely, the graph-theoretic Mertens theorems.

2. Prime zeta function of a graph

In this section, we give a proof of parts (1) and (2) of the Main Theorem introduced in Section 1. The following facts about Z X (u), etc., are known, and are often used in this paper. Facts 1. (1) (Basic facts) For an arbitrary real number t > 1, set u = t−s. Then the function ᐆX (s) is absolutely convergent and holomorphic for all s satisfying Re(s) > −log RX /log t (≥ 0). Since the function Z X (u) is the reciprocal of a polynomial by the Ihara–Bass formula, the function Z X (u) is meromorphic for all u ∈ ރ, and therefore ᐆX (s) is also meromorphic for all s ∈ ރ. (2) [Kotani and Sunada 2000, Theorem 1.3(1)] Let q +1 and p+1 be the maximum and minimum degrees of a graph X, respectively. Then 1/q ≤ RX ≤ 1/p, the point u = RX is a simple pole of Z X (u), and every pole of Z X (u) satisfies RX ≤ |u| ≤ 1. (3) [Terras 2011, p. 197] Suppose that X is a finite connected graph without degree- one vertices. Then RX = 1 if and only if X is a cycle graph. This follows from the equation p = q = 1.

(4) [Kotani and Sunada 2000, p. 8] The leading coefficient of the polynomial f X is given by Y c = (deg(v) − 1), v∈V

−ν and therefore that of the polynomial 1/Z X is equal to c2 = (−1) c. In this section, the following lemma is important. Key Lemma 2. Let d X i φ(u) = 1 + ci u ∈ ޚ[u] i=1 ONAPRIMEZETAFUNCTIONOFAGRAPH 129 be a polynomial function of degree d ≥ 0, and let T = {u ∈ ރ : there exists n ≥ 1 such that φ(un) = 0} denote the zeroes of the φ(un). Suppose that r is an arbitrary real number, and assume that 8(u) is a series defined by ∞ X 1 8(u) = log φ(un). nr n=1 Then 8(u) is absolutely convergent for u satisfying |u| < 1 and u ∈/ T. Proof. First, we suppose that d = 0. Then the φ(un) = 1 are constant, and therefore 8(u) = 0 is also constant. Hence, the claim is trivial. From now on, we assume that d ≥ 1. Set c := max{|ci | : 1 ≤ i ≤ d}, choose a number C0 with C0 ≥ cd + 1 (≥ 2), and fix it. Let rn (n ≥ 3) be a number defined by 1 − exp(−1/n2−r )1/n rn := . C0 1/n Note that rn < (1/C0) , the sequence {rn}n≥3 is increasing, and limn→∞ rn = 1. Take u satisfying |u| < 1 and u ∈/ T , and fix it. Then there exists a number N such that |u| ≤ rN , and thus |u| < rn for all n ≥ N + 1. Now we fix such numbers N and n. 1/n n Since |u| < (1/C0) and |u | ≤ |u| < 1, we obtain, by the triangle inequality, n n n n (1) 0 < 1 − C0|u | ≤ |φ(u )|, and so −log |φ(u )| ≤ −log(1 − C0|u |). n 2−r On the other hand, since |u| < rn, then C0|u | < 1 − exp(−1/n ), so we n 2−r obtain the inequality −log(1 − C0|u |) < 1/n . Combining this result with (1), we obtain 1 (2) Re(−log φ(un)) = −log |φ(un)| < . n2−r The first inequality in (1) also shows that the function log φ(un) is holomorphic in the closed disk |u| ≤ rN+1. By applying the Borel–Carathéodory theorem (see, e.g., n [Titchmarsh 1958, §5.5]) to the function log φ(u ) and the two circles |u| = rN+1, |u| = rN , we obtain 1 | φ(un)| ≤ | φ(un)| ≤ K (− φ(un)) ≤ K , log max log max Re log 2−r |u|=rN |u|=rN+1 n where K := 2rN /(rN+1 − rN ). Therefore, it follows that ∞ ∞ X 1 X 1 |log φ(un)| ≤ K < K · ζ(2) < ∞. nr n2 n=N+1 n=N+1 130 TAKEHIRO HASEGAWA AND SEIKENSAITO

Hence, for u satisfying |u| < 1 and u ∈/ T , the series 8(u) converges absolutely.  Using this lemma, we can prove the following proposition.

Proposition 3. Let µ(n) denote the Möbius function. If |u| < RX , then ∞ X µ(n) (3) P (u) = log Z (un). X n X n=1 Moreover, the right-hand side of (3) is absolutely convergent for u satisfying |u| < 1 and u ∈/ T , and therefore PX (u) extends analytically to the region {u ∈ ރ : |u| < 1}\T . Equivalently, if Re(s) > −log RX /log t, then ∞ X µ(n) (4) ᏼ (s) = log ᐆ (ns). X n X n=1 The right-hand side of (4) is absolutely convergent for s satisfying Re(s) > 0 and −s t ∈/ T , and so (4) gives the analytic continuation of ᏼX (s) to the region. Q∞ n −µ(n)/n Proof. Note that RX ≤ 1 (from Fact 1(2)) and exp(z) = n=1(1 − z ) for `(P) |z| < 1. Suppose that |u| < RX . Since |u | ≤ |u| < 1, we obtain the equality ∞ ∞ Y `(P) Y Y n`(P) −µ(n)/n Y n µ(n)/n exp(PX (u)) = exp(u ) = (1 − u ) = Z X (u ) , [P] [P] n=1 n=1 and therefore (3) holds for u satisfying |u| < RX . Set 2 −ν 2 1/Z X (u) = (1 − u ) f X (u) = 1 + c1u + · · · + c2u ∈ ޚ[x], c = max{|ci | : 1 ≤ i ≤ 2} and C0 = 2c ≥ 2. By applying Key Lemma 2 to φ(u) = 1/Z X (u) and r = 1, it follows that, for u satisfying |u| < 1 and u ∈/ T , P∞ n the series n=1 log Z X (u )/n is absolutely convergent, and so the right-hand side of (3) is absolutely convergent.  Moreover, for a Ramanujan graph, we can prove the following. Corollary 4. Suppose that X is a finite connected Ramanujan graph with degree q + 1, that is, Z X (u) satisfies the Riemann hypothesis (see Theorem 7.4 in[Terras 2011]). Then the function PX (u) is absolutely convergent for u satisfying |u| < 1 and |u| 6= (1/q)1/n for all n. Equivalently, the function ᏼX (s) is absolutely convergent for s such that Re(s)>0 and Re(s) 6= log q/ log tn for all n. Proof. Since X is a Ramanujan graph, by Theorem 1.3 in[Kotani and Sunada 2000], √ every real (resp. nonreal) zero of f X (u) satisfies |u| = 1 or 1/q (resp. |u| = 1/ q). 1/n n Thus, every point |u| 6= (1/q) is not zero of f X (u ). Hence, the proof of the assertion follows from Proposition 3.  ONAPRIMEZETAFUNCTIONOFAGRAPH 131

We can completely interchange the roles of the functions PX (u) and log Z X (u). Corollary 5. If |u| < 1 and u ∈/ T , then ∞ X 1 (5) log Z (u) = P (un). X n X n=1 Equivalently, if Re(s) > 0 and t−s ∈/ T , then ∞ X 1 (6) log ᐆ (s) = ᏼ (ns). X n X n=1 Proof. By applying the Möbius inversion formula (see, e.g., Theorem 270 in[Hardy and Wright 2008], or Theorem 2.2.8 in[Jameson 2003]) to the equality (3) for |u| < 1, we obtain the equality (5). 

Remark 6. The equalities (4) and (6) indicate that ᏼX (s) is a graph-theoretic analogue to the prime zeta function P(s) for the Riemann zeta function ζ(s). The relations between P(s) and ζ(s) are given as follows (see[Glaisher 1891], and also [Fröberg 1968] and Equality (1.6.1) in[Titchmarsh 1986]): For Re(s) > 1, ∞ ∞ X µ(n) X 1 P(s) = log ζ(ns) and log ζ(s) = P(ns). n n n=1 n=1 We can orient the edges of X, and label the edges as follows: E = { = −1 = −1 = −1} E a1, a2,..., a, a+1 a1 , a+2 a2 ,..., a2 a .

Let W = WX := (wi j ) denote the edge adjacency matrix of a graph X, that is, a 2 × 2 matrix defined by  −1 E 1 if t(ai ) = o(a j ) and a j 6= ai for ai , a j ∈ E, wi j := 0 otherwise

(see p. 28 in[Terras 2011]). Let λ1, . . . , λk be the distinct eigenvalues of W , and let e ,..., e be their multiplicities. Note that Pk e = 2. Let e := Pk e . 1 k i=1 i i=1,λi 6=±1 i By the determinant formula given by Hashimoto[1989] and Bass[1992], the polynomial 1/Z X (u) can be written as

k Y ei 1/Z X (u) = det(I2 − Wu) = (1 − λi u) . i=1

Note that f X (1) = 0. We now define a polynomial gX (u) by

gX (u) := f X (u)/(1 − u). 132 TAKEHIRO HASEGAWA AND SEIKENSAITO

0 Note that since f X (1) = 2( − ν)κ by[Northshield 1998, Theorem], 0 gX (1) = − f X (1) = −2( − ν)κ, where κ is the complexity of X, that is, the number of spanning trees in X. Since X is a non-cycle graph, that is,  6= ν, the polynomial gX (u) can be also written as k 1/Z (u) X 2ν−1−e Y ei (7) gX (u) = = (1 + u) (1 − λi u) . (1 − u2)−ν(1 − u) i=1 λi 6=±1

We can show that the function ᏼX (s) has a natural boundary. Proposition 7. Let X = (V, E) be a finite, connected and non-cycle graph without degree-one vertices. (1) There exists an eigenvalue λ of W such that |λ| > 1.

(2) The imaginary axis Re(s) = 0 is a natural boundary for the function ᏼX (s), that is, every point on this line can be realized as a limit point of singularities of ᏼX (s).

Proof. (1) The leading coefficient c2 of the polynomial 1/Z X (u) is given by k −ν Y Y ei (−1) (deg(v) − 1) = c2 = λi v∈V i=1 (from Fact 1(4)). By our assumption for X, the graph X is not a 2-regular graph. Thus |c2| > 1 and so there exists an eigenvalue λi with |λi | 6= 1. Note that every pole 1/λi of Z X (u) satisfies |1/λi | ≤ 1 by Fact 1(2). So there exists an eigenvalue λi with |λi | > 1. Q∞ n −µ(n)/n (2) Note that exp(z) = n=1(1 − z ) for |z| < 1. If |u| < 1 and u ∈/ T , then ∞ Y n µ(n)/n exp(PX (u)) = Z X (u ) n=1  ∞ −ν ∞  ∞ Y 2n −µ(n)/n Y n −µ(n)/n Y n −µ(n)/n = (1 − u ) (1 − u ) gX (u ) n=1 n=1 n=1 ∞ 2 Y n −µ(n)/n = exp(( − ν)u + u) gX (u ) , n=1 and therefore the equality ∞ X µ(n) P (u) = ( − ν)u2 + u − log g (un) X n X n=1 holds. ONAPRIMEZETAFUNCTIONOFAGRAPH 133

−s Note that u = t . By using the equalities (7) and2, the function ᏼX (s) can be written as

−2s −s ᏼX (s) = ( − ν)t + t ∞ k X µ(n) X  − (2ν − 1 − e) log(1 + t−ns) + e log(1 − λ t−ns) n i i n=1 i=1 λi 6=±1 for all s satisfying Re(s) > 0. By part (1), there exists λ such that |λ| > 1 −ns among the eigenvalues λ1, . . . , λk of W. Note that 1 − λt = 0 if and only if s = r(λ, n, m), where log |λ| Arg(λ) + 2πm r(λ, n, m) := + i , n log t n log t and Arg(λ) is the argument of λ with −π ≤ Arg(λ) < π. Note that log |λ| εn := → 0 n log t as n → ∞. We now fix an arbitrary point α = ia on the imaginary axis Re(s) = 0. Then, we can arrange a sequence of integers {mn} for each integer n so that Arg(λ) + 2πm n → a n log t as n → ∞. Hence, each point α on the boundary is a limit point of singularities of ᏼX (s). Since εn > 0 for all n, we cannot continue ᏼX (s) beyond the boundary at Re(s) = 0.  Remark 8. Proposition 7(2) is an analogue of the fact that the imaginary axis Re(s) = 0 is a natural boundary for the prime zeta function P(s) of the Riemann zeta function ζ(s) (see[Landau and Walfisz 1920]).

3. Graph-theoretic Mertens’ theorem

In this section, we prove parts (3)–(5) of the Main Theorem introduced in Section 1. Throughout this section, we always assume that X = (V, E) is a finite, connected, non-cycle graph without degree-one vertices. Note in particular that ν 6=  and 0 < RX < 1. First, we define the constants HX , CX and γX , and study their properties, which play important roles in this section. Let u be a complex variable. We define a function by X 1 X X 1 H (u) := log Z (u) − P (u) = P (un) = un`(P). X X X n X n n≥2 [P] n≥2 134 TAKEHIRO HASEGAWA AND SEIKENSAITO

Note that the point u = RX is a common pole of Z X (u) and PX (u) by Fact 1(2), and that the series HX (u) is absolutely convergent for u satisfying |u| < 1 and u ∈/ T , from Corollary 5. Since u = RX is a simple pole of Z X (u), we can define constants cX and CX by −1 cX := − Res Z X (u) = lim (RX − u)Z X (u) = u=R u↑R 2 −ν 0 X X (1 − RX ) f X (RX ) and CX := cX /RX .

Lemma 9. (1) The value HX := HX (RX ) is finite.

(2) The constants cX and CX are positive.

n Proof. (1) Since RX < RX < 1 (n ≥ 2), the function PX (u) is holomorphic at n n u = RX , and therefore PX (u ) is holomorphic at u = RX . We have

X X 1 n`(P) X X n`(P) H (R ) = R ≤ R X X n X X [P] n≥2 [P] n≥2 2`(P) 2 X R 1 X P PX (R ) = X ≤ R2`( ) = X < +∞, `(P) 1 − R X 1 − R [P] 1 − RX X [P] X and the assertion follows.

(2) Note that the leading coefficient of the polynomial f X is given by Y c = (deg(v) − 1) > 0 v∈V by Fact 1(4). Then f X factors as the product of irreducible polynomials such that

m m Y1 Y2 f X (u) = c (u − αi ) · f j (u), i=1 j=1 where the f j are monic of deg f j = 2, and deg f X = 2ν = m1 + 2m2. Note that m1 is even. Since u = RX is a simple pole of Z X (u), it is a simple zero of f X . We may assume that α1 = RX . Since αi > RX (2 ≤ i ≤ m1) and the discriminants of the f j are negative, the sign of

m1 m2 0 Y Y f X (RX ) = c (RX − αi ) f j (RX ) i=2 j=1

m1−1 0 is equal to (−1) = −1, i.e., f X (RX ) < 0, so cX > 0 and CX = cX /RX > 0.  ONAPRIMEZETAFUNCTIONOFAGRAPH 135

Since the function Z X (u) − cX /(RX − u) is holomorphic at u = RX , we can define a constant γX by   cX γX := lim Z X (u) − , u↑RX RX − u which is an analogue of the Euler–Mascheroni constant γ = lims↓1(ζ(s)−1/(s−1)) for ζ(s). In a neighborhood of u = RX , the function Z X (u) can be expanded as

cX Z X (u) = + γX + O(RX − u), RX − u and so

cX (8) log Z X (u) = log + O(RX − u). RX − u

Similarly, in a neighborhood of u = RX , the function PX (u) can be expanded as

cX cX PX (u) = log − HX (u)+O(RX −u) = log − HX (RX )+O(RX −u). RX − u RX − u In this section, the following facts are used. Facts 10. (1) (See, for example, Theorem 18.1 in[Korevaar 2002].) Let x be a P∞ n complex variable and let F(x) = n=0 an x be a power series with an ≥ 0 that converges for |x| < 1. Suppose that C F(x) − = O(1) 1−x P as x → 1. Then the partial sum A(N) = n≤N an satisfies A(N) = C · N + O(log N) as N → ∞. (2) (See, for example, Exercises 9-6 in[Apostol 1974], and Theorem 1.3.6 in [Jameson 2003], the Abel partial summation formula). Let {an} be real numbers, and let f (t) be a (real- or complex-valued) function with a continuous derivative in the interval [1, N]. Then Z N X 0 an f (n) = A(N) f (N) − A(t) f (t) dt. n≤N 1 By using Fact 10, we can prove the following proposition. Proposition 11. Suppose that X is a finite, connected and non-cycle graph with- out degree-one vertices. In a neighborhood of u = RX , expand Z X (u) into the 136 TAKEHIRO HASEGAWA AND SEIKENSAITO power series ∞ X 0 n Z X (u) = anu . n=0 Then, as N → ∞, X 0 n an RX = CX · N + O(log N). n≤N

Proof. First, for simplicity of arguments, we normalize the function Z X (u):

∞ ∞ X 0 n n X n F(x) = Z X (RX x) = an RX x = an x , n=0 n=0

0 n where an = an RX . Note that the normalized function F(x) converges for |x| < 1. 0 Since all coefficients an are nonnegative (by page 13 in[Terras 2011]), all coeffi- cients an are also nonnegative. Since X is a non-cycle graph, the point x = 1 is a simple pole of F(x). Hence, we obtain C F(x) − X = O(1) 1 − x as x → 1. By applying Fact 10(1) to this equality, as N → ∞,

X X 0 n an = CX · N + O(log N), and so an RX = CX · N + O(log N) n≤N n≤N holds, and the assertion follows.  Now, we compute the following example.

Example 12 [Terras 2011, Example 2.8, p. 18]. Consider the graph X = K4 − {one edge}. Then

2 2 2 3 −1 2 f X (u)=(1−u)(1+u )(1+u+2u )(1−u −2u ) and Z X (u) =(1−u ) f X (u).

Since the radius of convergence RX of Z X (u) is the smallest positive real zero of f X (u), √ = 1 − + −1 = = + 1/3 RX 6 (α 1 α ) 0.6572981 . . . , α (53 6 78) .

Then CX is computed as CX = 0.5540954 ... . For example, if N = 50000, then

1 X a0 Rn = 0.5540867 ... ≈ C . N n X X n≤N ONAPRIMEZETAFUNCTIONOFAGRAPH 137

Let X = (V, E) be a graph, and set |V | = ν and |E| = . Let W = WX be the edge adjacency matrix of X (see page 28 in[Terras 2011], or Section 2 in this paper), and let Spec(W) denote the spectrum of W, that is, the list of its eigenvalues together with their multiplicities. Note that |Spec(W)| = 2. The polynomial 1/Z X (u) has an expression different from that in Section 2. In fact, this can be written as

 k  Y Y ei 1/Z X (u) = det(I2 − Wu) = (1 − λu) = (1 − λi u) . λ∈Spec(W) i=1

Since the points u = 1/λ are the poles of Z X (u), we obtain 1 ≤ |λ| ≤ 1/RX by Fact 1(2). The following lemma is used in the proof of Theorem 14 in this section.

Key Lemma 13. Suppose that X is a finite, connected and non-cycle graph without degree-one vertices.

(1) As N → ∞, we have

N X X n (λRX ) = N + O(1). n=1 λ∈Spec(W)

1 (2) Let 0 < α < 2 be a fixed real number. Then there exists a natural number N0 such that, for any n ≥ N0,

(1−α)n X n  1  n · π(n) − λ < 2 . RX λ∈Spec(W)

Proof. (1) Let 1X denote

1 = 1X := gcd{`(P) :[P] is a prime in X}

(see Definition 2.12 in[Terras 2011]). It follows from Theorem 1.4 in[Kotani and Sunada 2000] that the poles of Z X (u) on the circle |u| = RX have the form 2πia/1 u = RX e (1 ≤ a ≤ 1). It is well known that

1 X 1 if 1 | n, e2πian/1 = 0 otherwise a=1 (see, e.g., Exercise 10.1 in[Terras 2011]). Then we obtain

N N 1 X X n X X 2πian/1 h N i N − (λRX ) = N − e = N − 1 < 1, 1 |λ|=1/RX n=1 n=1 a=1 138 TAKEHIRO HASEGAWA AND SEIKENSAITO where [r] denotes the integer part of the real number r. On the other hand, we obtain N X X n X n 2ρ RX (λRX ) < 2 (ρ RX ) = , 1 − ρ RX |λ|<1/RX n=1 n≥1 where

ρ := max{|λ| : λ ∈ Spec(W), |λ| < 1/RX }. Combining these inequalities, by the triangle inequality we obtain N X X n 2ρ RX N − (λRX ) < 1 + 1 − ρ RX n=1 λ∈Spec(W) as N → ∞, and the assertion follows. P (2) Let µ(n) denote the Möbius function. Note that d|n |µ(d)| ≤ n. It is known that 1 X X π(n) = µ(d)N and N = λn n n/d n d|n λ∈Spec(W) (see (10.3) and (10.4) in[Terras 2011]). Combining these equalities, we obtain X X n · π(n) = µ(d)λn/d , λ∈Spec(W) d|n and thus

X n X X n/d n · π(n) − λ = µ(d)λ

λ∈Spec(W) λ∈Spec(W) d|n d≥2 X X X X ≤ |µ(d)| · |λ|n/d ≤ |µ(d)| · |λ|n/2 λ∈Spec(W) d|n λ∈Spec(W) d|n d≥2 d≥2 X  1 n/2  1 n/2 ≤ n ≤ 2n . RX RX λ∈Spec(W)

1 On the other hand, since RX < 1 and 0 < α < 2 by our assumptions, there exists a natural number N0 such that, for any n ≥ N0,  1 (1/2−α)n  1 n/2  1 (1−α)n n ≤ , and so n ≤ . RX RX RX

Hence, for any n ≥ N0,

(1−α)n X n  1  n · π(n) − λ ≤ 2 , RX λ∈Spec(W) ONAPRIMEZETAFUNCTIONOFAGRAPH 139 and the assertion follows.  At last, we can prove the main theorem in this section.

Theorem 14. Suppose that X is a finite, connected and non-cycle graph without degree-one vertices. Let γ = 0.57721 ... be the Euler–Mascheroni constant, and let HX = HX (RX ) and CX be the constants.

(1)( Graph-theoretic Mertens’ first theorem) As N → ∞,

X n n · π(n)RX = N + O(1). n≤N

(2) (Graph-theoretic Mertens’ second theorem) There exists a constant BX such that, as N → ∞, X  1  π(n)Rn = log N + B + O . X X N n≤N

(3) The equality BX = γ + log CX − HX holds. Equivalently,

X X 1 n`(P) B = γ + log C − R X X n X [P] n≥2 Y `(P) `(P)
= γ + log CX + log(1 − RX ) + RX . [P]

(4)( Graph-theoretic Mertens’ third theorem) As N → ∞,

−γ Y `(P)
Y n π(n) e 1 1 − RX = (1 − RX ) ∼ · . CX N `(P)≤N n≤N

Proof. (1) Let N0 be a number as in the proof of Key Lemma 13(2), and let K denote the constant

N0−1 N0−1 X n X X n K := n · π(n)R − (λRX ) . X n=1 n=1 λ∈Spec(W)

Assume that N is sufficiently large. Then it follows from Key Lemma 13(2) that

N N N   X n X X n X n X n n · π(n)R − (λRX ) ≤ K + R n · π(n) − λ X X n=1 n=1 λ∈Spec(W) n=N0 λ∈Spec(W) N 2 ≤ + X αn + K 2 RX < K α , 1 − RX n=N0 140 TAKEHIRO HASEGAWA AND SEIKENSAITO and therefore by Key Lemma 13(1) we have

N N X n X X n n · π(n)RX = (λRX ) + O(1) = N + O(1) as N → ∞. n=1 n=1 λ∈Spec(W)

n (2) We set an = n · π(n)RX . By part (1), we obtain A(t) = t + O(1). By applying Fact 10(2) with f (t) = 1/t, we get

X A(N) Z N A(t) N + O(1) Z N t + O(1) π(n)Rn = + dt = + dt X N t2 N t2 n≤N 1 1   Z N    = + 1 + 1 + 1 1 O O 2 dt N 1 t t  1  h 1iN = 1 + O + log t + O N t 1  1   1   1  = 1 + O + log N + O + O(1) = log N + O(1) + O , N N N and the assertion follows.

n (3) Fix an arbitrary x satisfying 0< x <1. By applying Fact 10(2) with an =π(n)RX and f (t) = xt ,

Z N X n n N t π(n)RX x = A(N)x − log x x A(t) dt n≤N 1 holds. It follows from part (2) that

X   1  Z N  1 π(n)Rn xn = log N +B +O x N −log x xt log t +B +O dt, X X N X t n≤N 1 and, moreover, as N → ∞,

Z ∞ t  1 (9) PX (RX x) = −log x x log t + BX + O dt. 1 t

In order to calculate the right-hand side of this equality, for simplicity of argu- ments, we define the functions In = In(x):

Z ∞ t  1 −log x x log t + BX + O dt = I1 + I2 + O(I3), 1 t ONAPRIMEZETAFUNCTIONOFAGRAPH 141 where Z ∞ t I1 = −log x x log t dt, 1 Z ∞ t I2 = −BX · log x x dt = BX · x, and 1 ∞ Z xt I3 = −log x dt. 1 t

First, we compute the function I1: Z ∞ Z ∞ t t 0 x I1 = − (x ) log t dt = dt. 1 1 t Now we take r = −t log x. Note that log x < 0. Then we obtain ∞ Z e−r I1 = dr = −Ei(log x), −log x r where Ei(z) (z ∈ ރ and |Arg(−z)| < π) is the exponential integral ∞ Z e−r −Ei(−z) = dr z r (see, e.g., Equality (3.1.3) in[Lebedev 1972]). Since the function Ei(z) expands as ∞ X zk Ei(z) = γ + log(−z) + k · k! k=1 (see Equality (3.1.6) in [ibid.]),

I1 = −γ − log(−log x) + O(log x) = −γ − log(−log x) + O(1 − x).

Next we calculate the function I3. It follows from the above result that ∞ Z xt I3 = −log x dt = (−log x)I1 = O(1 − x) 1 t as x ↑ 1. By combining the above results, the equality (9) is written as follows:

PX (RX x) = −γ − log(−log x) + BX x + O(1 − x), and, moreover, as x ↑ 1,

(10) PX (RX x) + log(−log x) → BX − γ. On the other hand, since 1 log Z (R x) = log + log C + O(1 − x) X X 1−x X 142 TAKEHIRO HASEGAWA AND SEIKENSAITO from the equality (8), as x ↑ 1,

−log x  (11) log Z X (RX x) + log(−log x) = log + log CX → log CX . 1 − x Combining (10) with (11), we obtain
HX = lim HX (RX x) = lim log Z X (RX x) − PX (RX x) x↑1 x↑1
= lim (log Z X (RX x) + log(−log x)) − (PX (RX x) + log(−log x)) x↑1

= log CX + γ − BX .

(4) Fix an arbitrary positive real number N. We define the following functions:

∞ ∞ ≤ X X 1 X X 1 H N = π(n) Rmn and H >N = π(n) Rmn. X m X X m X n≤N m=2 n>N m=2

≤N >N Note that HX = HX + HX . From parts (2) and (3), we obtain

X ≤  1  π(n)Rn + H N = log N + γ + log C − H >N + O . X X X X N n≤N

Since the left-hand side of this equality is equal to

∞ X ≤ X X 1 π(n)Rn + H N = π(n) Rmn X X m X n≤N n≤N m=1   X n Y n π(n) = − π(n) log(1 − RX ) = − log (1 − RX ) , n≤N n≤N we obtain

−γ Y n π(n) e 1  >N  1  (1 − RX ) = · exp HX + O . CX N N n≤N

>N Since HX → 0 and 1/N → 0 as N → ∞, the assertion follows.  Last, we compute the following example.

Example 15 (continued from Example 12). Consider the graph X=K4−{one edge}. Then

HX = 0.25613 ..., BX = γ + log CX − HX = −0.26933 .... ONAPRIMEZETAFUNCTIONOFAGRAPH 143

For example, if N = 550, then

X n π(n)RX − log N = −0.26842 · · · ≈ BX , n≤N − Y e γ 1 (1 − Rn )π(n) = 0.18447 · · · ≈ · = 0.18457 .... X C N n≤N X Remark 16. (See[Mertens 1874, Equation (17)], or[Hardy and Wright 2008, Theorem 428].) A number-theoretic analogue to part (3) in the preceding theorem is X  1  1  B = γ − H = γ + log 1 − + , 1 p p p P where H = n≥2 P(n)/n is a constant, and P(s) is the prime zeta function. Remark 17. We now compare parts (2)–(4) of our Theorem 14 with Theorem 1 in [Sharp 1991]. We define

h X `(P) h X N h X := −log RX , N(P) = e and x = e .

The quantity h X is called the topological entropy of a flow in ergodic theory (see [Sharp 1991]), which is a constant in our setting. Note that `(P) ≤ N if and only if `(P) N(P) ≤ x. Note that RX = 1/N(P). Then our Mertens’ second theorem can be rewritten as X 1  1  = log(log x) + B + O , N(P) log x N(P)≤x where B := −log h X + BX , and, similarly, our Mertens’ third theorem becomes − Y  1  1 e γ 1 − ∼ · . N(P) CX /h X log x N(P)≤x

In Theorem 1 in[Sharp 1991], our constant CX /h X , which is equal to a residue (up to sign) of the Ihara zeta function, corresponds with that of a dynamical zeta function for a flow. Moreover, our Theorem 14(3) becomes X  1  1  B = γ + log(C /h ) + log 1 − + . X X N(P) N(P) [P] Remark 18. Let X = (V, E) be a finite, connected, non-cycle graph without degree- one vertices, and let S = (V 0, E0) be its k-subdivision (that is, let S be the graph obtained from X by adding k new vertices to each edge of X) (see Examples 6.4 and 8.5 in[Terras 2011]). Then

HX = HS, CX = (k + 1)CS, and BX = BS + log(k + 1). 144 TAKEHIRO HASEGAWA AND SEIKENSAITO

k+1 This is proved as follows: note that 1S = (k + 1)1X , RS = RX , and  πX (n/(k + 1)) if (k + 1) | n, πS(n) = 0 otherwise. Therefore, ∞ ∞ X 1 X 1 X X 1 X (k+1)mn H = P (Rm) = π (n)Rmn = π (n)R S m S S m S S m X S m≥2 m≥2 n=1 m≥2 n=1 ∞ X 1 X X 1 = π (n)Rmn = P (Rm) = H . m X X m X X X m≥2 n=1 m≥2

0 0 k+1 Note that ν = ν + k,  = (k + 1), and Z S(u) = Z X (u ), and so

2 −ν 2(k+1) −ν k+1 (1 − u ) fS(u) = (1 − u ) f X (u ), 2 −ν 0 2 −ν 0 (1 − RS) RS fS(RS) = (k + 1)(1 − RX ) RX f X (RX ). Therefore, −(k + 1) (k + 1)C = = C , S 2 0−ν0 0 X (1 − RS) RS fS(RS) and so

BX = γ + log CX − HX = γ + log CS − HS + log(k + 1) = BS + log(k + 1).

Acknowledgements

Saito would like to thank Professor Toyokazu Hiramatsu for his encouragement and thoughtful suggestions. The authors thank the referee for his or her valuable comments and careful review of this paper.

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Received October 3, 2013. Revised May 29, 2014. 146 TAKEHIRO HASEGAWA AND SEIKENSAITO

TAKEHIRO HASEGAWA SHIGA UNIVERSITY OTSU SHIGA 520-0862 JAPAN [email protected]

SEIKEN SAITO WASEDA UNIVERSITY SHINJUKU TOKYO 169-8050 JAPAN [email protected] PACIFIC JOURNAL OF MATHEMATICS msp.org/pjm

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